The finite-difference time-domain (FDTD) method requires a minimum of ten cells per wavelength to assume a linear field transition between adjacent cells in order to consistently achieve accuracy. Maintaining this relationship for good conductors is not practical because the fields are decaying exponentially over the skin depth, which is a short distance when compared to the free space wavelength. The surface conductivity correction in XF can be enabled for good conductors to more accurately model the losses and reflections at the boundary of a metal.

## Background

FDTD's discrete representation of Maxwell's equations creates a need for XF's surface connectivity correction. In order to accurately model field interactions at metal boundaries, the conductivity needs to be modified so the impedance used in FDTD matches the theoretical value.

In this case, assume a time-harmonic transverse electromagnetic wave propagating in lossy media. Further, simplify the wave to traveling in the z direction and containing only the x component of the electric field.

For the theoretical case, the equation for the electric field is written as

$${\bf E}(z) = \hat a_xE_x(z) = \hat a_x\left(E_0^+e^{-\gamma z}+E_0^-e^{+\gamma z}\right)$$

where the two terms represent the positive and negative traveling waves. $\gamma$ is the propagation constant and takes the form

$$\gamma=\sqrt{-\omega^2\mu\epsilon+j\omega\mu\sigma_t}$$

$\sigma_t$ is the theoretical conductivity of the material. The corresponding magnetic field is determined from Maxwell's equations as

\begin{eqnarray} {\bf H}&=&-\frac{1}{j\omega\mu}\nabla\times{\bf E} \\ &=&-\hat a_y\frac{1}{j\omega\mu}\frac{\partial E_x}{\partial z} \\ &=&\hat a_y\frac{\gamma}{j\omega\mu}\left(E_0^+e^{-\gamma z}-E_0^-e^{+\gamma z}\right) \\ &=&\hat a_y\sqrt{\frac{\sigma_t+j\omega\epsilon}{j\omega\mu}}\left(E_0^+e^{-\gamma z}-E_0^-e^{+\gamma z}\right) \end{eqnarray}

From the above equation, the theoretical wave impedance, or ratio of electric to magnetic field, is represented as

$$Z_t=\sqrt{\frac{j\omega\mu}{\sigma_t+j\omega\epsilon}}=\eta_t$$

This is equivalent to the intrinsic impedance, $\eta_t$ of the lossy medium.

FDTD approximates the partial derivatives in Maxwell's equation by discretizing them, causing the representation of the material's intrinsic impedance to be different. Starting with Maxwell's equation, the z directed wave is represented

$$\frac{\partial H_y}{\partial z} = \epsilon \frac{\partial E_x}{\partial t} + \sigma_f E_x$$

where $\sigma_f$ is the conductivity of the material in FDTD. In partial FDTD notation, Maxwell's equation takes the form

$$\frac{\left(H_y\big|_{i,j,k+1/2}-H_y\big|_{i,j,k-1/2}\right)}{\Delta z} = \epsilon\frac{E_x\big|^{n-1/2}_{i,j,k} - E_x\big|^{n+1/2}_{i,j,k}}{\Delta t} + \sigma_f E_x\big|_{i,j,k}$$

where $i,j,k$ represent the spatial index of the fields. Notice the use of $z-1/2$ because the magnetic fields are separated from the electric fields in space by a half cell.

Rearranging the terms and simplifying the equation with assumptions for a good conductor allows the relation between the electric and magnetic fields to be written as

$$E_x\big|_{i,j,k}\simeq\frac{\Delta t}{(\epsilon+\sigma_f\Delta t)}\left[\frac{H_y\big|_{i,j,k+1/2}}{\Delta z}\right]$$

Since $\sigma\Delta t\gg\epsilon$ for good conductors, the intrinsic impedance in FDTD is written

$$\eta_f\simeq\frac{1}{\sigma_f\Delta z}$$

In order to accurately model reflections and losses at good conductor boundaries, XF's surface conductivity correction setting is necessary for matching the impedance used in FDTD to that from theory, $\eta_f = \eta_t$.

$$\frac{1}{\sigma_f\Delta z} = \sqrt{\frac{j\omega\mu}{\sigma_t+j\omega\epsilon}}$$

Since FDTD is a time-domain formulation, the impedance can be maintained by solving for the real part of $\sigma_f$.

$$\sigma_f=\frac{1}{\Delta z}\sqrt{\frac{2\sigma_t}{\omega\mu}} \label{eq:sigmaf}$$

The surface conductivity correction setting adjusts the material conductivity in the calculation engine to $\sigma_f$ from the theoretical value, $\sigma_t$, specified by the user in the user interface.

## Material Editor

For isotropic, nondispersive materials with a normal entry method, the Material Editor provides a Surface Conductivity Correction setting, which should be checked when defining a highly-conductive metal. When using this option, the theoretical values for both conductivity, $\sigma_t$, and Relative Permittivity should be entered. The correction is applied at a single Evaluation Frequency that determines $\omega$, which is then used by the calculation engine to solve for $\sigma_f$ and maintain impedance.